EXPLICIT QUADRATIC CHABAUTY OVER NUMBER FIELDS JENNIFER S. BALAKRISHNAN, AMNON BESSER, FRANCESCA BIANCHI, AND J. STEFFEN MULLER¨ Abstract. We generalize the explicit quadratic Chabauty techniques for inte- gral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved by combining equations coming from Siksek’s extension of classical Chabauty with equations defined in terms of p-adic heights attached to inde- pendent continuous idele class characters. We give several examples to show the practicality of our methods. Contents 1. Introduction 1 Acknowledgements 5 1.1. Notation 6 2. p-adic heights 6 2.1. Continuous idele class characters 6 2.2. Coleman-Gross p-adic height pairings 8 3. Chabauty over number fields 10 4. Quadratic Chabauty for integral points on odd degree hyperelliptic curves over number fields 12 5. Quadratic Chabauty for rational points on bielliptic curves over number fields 16 6. Algorithms and examples for integral points over quadratic fields 20 6.1. Elliptic curves over real quadratic fields 21 6.2. Elliptic curves over imaginary quadratic fields 23 6.3. Genus 2 curves, imaginary quadratic fields 24 7. Examples for rational points 25 arXiv:1910.04653v2 [math.NT] 15 Jun 2020 7.1. Rank 4 over imaginary quadratic fields 25 7.2. Rank 3 over real quadratic fields 27 Appendix A. Roots of multivariate systems of p-adic equations 29 References 33 1. Introduction Let K be a number field and let X/K be a smooth projective curve of genus g 2. Then, by Faltings’ theorem, the set of rational points X(K) is finite, but at present≥ Date: June 16, 2020. 2010 Mathematics Subject Classification. Primary 11G30; Secondary 11S80, 11Y50, 14G40. 1 2 J.S.BALAKRISHNAN,A.BESSER,F.BIANCHI, AND J.S.MULLER¨ no general algorithm for the computation of X(K) is known. When the Mordell- Weil rank r of the Jacobian J/K is less than g, the method of Chabauty [Cha41], made effective by Coleman [Col85, MP12], can often be used to compute a finite subset of X(Kp) containing X(K), for any prime p of good reduction for X. Com- bined with other techniques such as the Mordell-Weil sieve [BS10], this suffices to compute X(K) exactly in many situations. In principle, this method can be used also to compute rational points on elliptic curves, but it only applies to the trivial case r = 0. The method of Chabauty-Coleman relies on the image of the map 0 (1.1) log: J(K) Kp H (X , Ω )∨ ⊗ → Kp 1 having positive codimension, which is used to write down abelian integrals vanishing at the rational points. Kim [Kim05, Kim09] proposed to extend this approach to curves with r g, by replacing the Jacobian with non-abelian Selmer varieties and the abelian≥ integrals with iterated Coleman integrals. This requires techniques from p-adic Hodge theory and, in general, appears to be quite difficult to use for computations of X(K). However, the simplest non-abelian instance of Kim’s program has been made explicit in various circumstances. For now suppose that K = Q. The basic idea is to use a p-adic height function, which is a quadratic form h: J(Q) Q . We can decompose h into a sum h = → p h of local terms h : Div0(X ) Q , where v runs through all prime numbers v v v Qv → p and h (D) = 0 for all but finitely many v if D Div0(X ) (see [CG89, BB15]). P v Q If (1.1) is injective, then we can write down a basis∈ of the quadratic forms on J(Q) Q in terms of products of abelian integrals, andQ we can write h as a linear ⊗ p combination h = q αqq over Qp. Hence we find ∈Q (1.2) P α q h = h . q − p v q v=p X∈Q X6 The left hand side can be pulled back to X(Q) and extended to a locally analytic function on a subset X(Q ). If the pull-back of the right hand side vanishes Sp ⊂ p on v=p v X(Q) for some v X(Qv) or, more generally, if we can control 6 S ∩ S ⊂ the image, then we get a locally analytic function on p which takes values on Q S v v X(Q) in an explicitly computable finite subset of Qp. OneS ∩ context in which this idea has been made explicit is the one of integral pointsQ on certain hyperelliptic curves. Suppose that X is hyperelliptic, given by a model y2 = f(x), where f Z[x] is monic of odd degree and has no repeated roots. It is shown in [BBM16] that∈ (1) the function τ(z) := hp(z ) is a Coleman function on X(Qp) , which can be expressed as a double−∞ Coleman integral ([BBM16, Theorem\{∞} 2.2]); (2) for v = p, the function z h (z ) takes values on the v-integral 6 7→ v − ∞ points of X(Qv) in an explicitly computable finite set Tv Qp ([BBM16, Proposition 3.3]), which is the set 0 for almost all v. ⊂ { } These results do not use Kim’s approach directly; the first uses p-adic Arakelov theory whereas the second is based on intersection theory on arithmetic surfaces. Let = Spec Z[x, y]/(y2 f(x)). Together with the discussion above, it is then easyU to deduce − Theorem 1.1 ([BBM16, Theorem 3.1]). Suppose that r = g and that the map in (1.1) is an isomorphism. Then there exists an explicitly computable finite set T ⊂ EXPLICIT QUADRATIC CHABAUTY OVER NUMBER FIELDS 3 Qp and an explicitly computable non-constant Coleman function ρ: (Zp) Qp such that ρ( (Z)) T . U → U ⊂ Here and in the following we write that certain p-adic objects are explicitly computable if we can provably compute them to any desired p-adic precision; see Remark 6.1. The first goal of the present article is to extend Theorem 1.1 to general number fields. Fix a prime number p such that X has good reduction at all p p and fix | a nontrivial continuous idele class character χ: AK∗ /K∗ Qp. This choice induces χ → χ Q -linear maps t : Kp Q for all p p and a p-adic height pairing h with values p p → p | in Qp. This can be decomposed into a sum of local height pairings and the local χ χ heights h are of the form t τp for some Coleman functions τp. Following a similar p p ◦ extension of classical Chabauty due to Siksek [Sik13], we work in X(K Q ) rather ⊗ p than in X(Kp) for a single place p p, and we consider the composition | (1.3) log: J(K) Q Res (J)(Q) Q Lie(Res (J)) . ⊗ p → K/Q ⊗ p → K/Q Qp Let σ : X(K) ֒ X(K Q ) denote the embedding induced by the product of the → ⊗ p .completion maps K ֒ K Qp = p p Kp → ⊗ | Theorem 1.2. Suppose that (1.3)Qis injective. Then there exists an explicitly com- χ putable finite set T Qp and an explicitly computable non-constant locally analytic χ ⊂ χ function ρ : ( K Zp) Qp, both dependent on χ, such that ρ (σ( ( K ))) T χ. U O ⊗ → U O ⊂ Remark 1.3. One can obtain a slight practical improvement of the theorem by noting that the sets T χ arise, just as in [BBM16], from possible values of the local χ components hv , and these values are highly dependent for varying χ because they are just the products of a certain intersection pairing with a constant depending on χv (see (2.6)). Thus, picking a basis χi for the space of idele class characters { } χ1 one may prove that the vector valued function ρ = (ρ ,...) takes on σ( ( K )) a finite computable set of possible values, which is smaller than the obviousU productO of the T χi . For simplicity we ignore this point here. See Theorem 4.2 below and its proof. Note that both Theorem 1.1 and Theo- rem 1.2 hold true when g = 1. The method of [BBM16] was generalized by the first author and Dogra in [BD18] to rational points on smooth projective curves, satisfying some additional condi- tions. In contrast to the proof of Theorem 1.1 in [BBM16], this generalization uses Kim’s approach directly, relating certain Selmer varieties to p-adic heights as constructed by Nekov´aˇr[Nek93], via p-adic Hodge theory. In a special case, this was turned into an explicit algorithm to compute a finite set of p-adic points containing the rational points of X. In particular, let X be a bielliptic curve of genus 2 over a number field K. Then X admits degree 2 maps ϕ1, ϕ2 to two elliptic curves E1 and E2, respectively. Theorem 1.4 ([BD18, Theorem 1.4]). Suppose that K is Q or an imaginary qua- dratic field in which p splits, and that E1 and E2 each have rank 1 over K; let p p. Then there exist a nontrivial continuous idele class character χ, an explicitly | χ computable finite set T Qp and an explicitly computable non-constant Coleman χ ⊂ χ χ function ρ : X(Kp) Q , such that ρ (X(K)) T . → p ⊂ 4 J.S.BALAKRISHNAN,A.BESSER,F.BIANCHI, AND J.S.MULLER¨ This explicit result from [BD18] can alternatively be proved only from properties of the local heights on the elliptic curves Ei and Theorem 1.1 for each Ei.